New York City second-quarter home sales were nearly cut in half from one year ago, but prices and even new building permits went up, according to a study from New York University Furman Center for real estate and urban policy. Home and condominium sales totaled 4,200 in the second quarter, down 40% from the same period last year and a 20% drop from the previous three months. The National Association of Realtors said the overall U.S. housing market started the third quarter with a steep drop in existing home sales, which could mean the New York City market could slow even further. Other indicators in the market pointed up. Home prices citywide increased 6% from the first quarter but remains 21% down from the peak. Prices increased 8% in the Bronx and Brooklyn from the previous quarter but remain more than 20% down from their peaks, according to the study. The average price for a single-family unit in New York City went for $415,000 in the second quarter and more than $5.3 million in Manhattan. More than 1,500 new unities were authorized for residential building permits in the first half of the year, nearly reaching the 1,700 for all of 2010. Foreclosures slipped from last year as well, according to the study. “We saw fewer foreclosure notices in this quarter than we did in the same quarter of 2010,” said Ingrid Gould Ellen, faculty co-director of the Furman Center. “Yet, nearly 7,000 households were newly affected by a foreclosure notice in the second quarter of 2011 and nearly 3,100 of those households were in Brooklyn.” Write to Jon Prior. Follow him on Twitter @JonAPrior